Hyponormality of Toeplitz Operators

Transcription

1 Hyponormality of Toeplitz Operators Carl C. Cowen Proc. Amer. Math. Soc. 103(1988) Abstract For ϕ in L ( D), let ϕ = f + g where f and g are in H 2.Inthis note, it is shown that the Toeplitz operator T ϕ is hyponormal if and only if g = c + T h f for some constant c and some function h in H ( D) with h 1. For ϕ in L ( D), the Toeplitz operator T ϕ is the operator on H 2 of the unit disk D given by T ϕ u = Pϕu where P is the orthogonal projection of L 2 ( D) ontoh 2.AnoperatorA is called hyponormal if its self-commutator A A AA is positive. The goal of this paper is to characterize hyponormal Toeplitz operators. Brown and Halmos began the systematic study of the algebraic properties of Toeplitz operators and showed, [3, page 98], that T ϕ is normal if and only if ϕ = α + βρ where α and β are complex numbers and ρ is a real valued function in L. There are many results concerning hyponormality of Toeplitz operators in the literature and properties of hyponormal Toeplitz operators have played an important role in work on Halmos s Problem 5, [7], Is every subnormal Toeplitz operator either normal or analytic? but a characterization has been lacking. (For references, see the bibliography; [6] surveys much of the literature.) Mathematics Subject Classification (1985 Revision). Primary 47B35, 47B20. 0 Key Words and Phrases. Toeplitz operator, subnormal operator. Supported in part by National Science Foundation Grant DMS

2 Theorem 1 If ϕ is in L ( D), whereϕ = f + g for f and g in H 2,then T ϕ is hyponormal if and only if g = c + T h f. for some constant c and some function h in H ( D) with h 1. The basis of the proof is a dilation theorem; we will use the notation and formulation of Sarason [13, Theorem 1]. The unilateral (forward) shift on H 2 will be denoted by U. Moreover, the proof uses standard results about Hankel operators, for example, see [12]. For ψ in L, the Hankel operator H ψ is the operator on H 2 given by H ψ u = J(I P )(ψu) where J is the unitary operator from H 2 onto H 2 J(e inθ )=e i(n 1)θ. Denoting by v the function v (e iθ )=v(e iθ ), another way to put this is that H ψ is the operator on H 2 defined by <zuv, ψ> = <H ψ u, v >, for all v H. (1) Necessary facts about Hankel operators include H ψ1 = H ψ2 if and only if (I P )ψ 1 =(I P )ψ 2. H ψ =inf{ ϕ :(I P )ψ =(I P )ϕ}. H ψ = H ψ. Either H ψ is one-to-one or ker(h ψ )=χh 2 where χ is an inner function. The closure of the range of H ψ is H 2 in the former case and (χ H 2 ) in the latter. H ψ U = U H ψ. Proof. Let ϕ = f + g where f and g are in H 2. 2

3 The first step of the proof is one of the equivalences of Proposition 11 of [6]. For every polynomial p in H 2, <(T ϕ T ϕ T ϕ T ϕ )(p),p> = <T ϕ p, T ϕ p> <T ϕ p, T ϕ p> = <f p + P ḡp,fp + P ḡp> <P fp+ gp, P fp+ gp> = < fp, fp> <P fp,p fp> <ḡp, ḡp> + <P ḡp,pḡp> = < fp,(i P ) fp> <ḡp, (I P )ḡp> = <(I P ) fp,(i P ) fp> <(I P )ḡp,(i P )ḡp> = H f p 2 Hḡp 2. Since the polynomials are dense in H 2 and since the Hankel and Toeplitz operators involved are bounded, we see that T ϕ is hyponormal if and only if for all u in H 2, Hḡu H fu. (2) Let K denote the closure of the range of H f, and let S denote the compression of U to K. SinceK is invariant for U, the operator S is the restriction of U to K. Suppose first that T ϕ is hyponormal. Define an operator A on the range of H f by A(H fu) =Hḡu. If H fu 1 = H fu 2,sothatH f(u 1 u 2 ) = 0, then the inequality (2) implies that Hḡ(u 1 u 2 ) = 0 too and it follows that A is well defined. Moreover, inequality (2) implies A 1soA has an extension to K, which will also be denoted A, with the same norm. Now by the intertwining formula for Hankel operators and the fact that K is invariant for U,wehave and also HḡU = AH fu = AU H f = AS H f HḡU = U Hḡ = U AH f = S AH f. Since the range of H f is dense in K, we find that AS = S A on K, or taking adjoints, that SA = A S. By [13, Theorem 1] (or by the usual theory of the unilateral shift if K = H 2 ), there is a function k in H ( D) with k = A = A such that A is 3

4 the compression to K of T k. Since K is invariant for T k that A is the restriction of T k to K and = T k, this means Hḡ = T k H f. (3) Conversely, if equation (3) holds for some k in H ( D) with k 1, then clearly inequality (2) holds for all u, andt ϕ is hyponormal. The proof will be completed by analyzing the relationship given by equation (3). Using the formulation (1), equation (3) holds if and only if for all H functions u, v, <zuv, g> = <H ḡu, v > = <T k H fu, v > = <H fu, kv > = <z uk v, f> = <zuv, k f> = <zuv, T k f>. Since the closed span of {zuv : u, v H } is zh 2 this means that equation (3) holds if and only if g = c + T h f for h = k. (Note that k = k.) In the cases for which T ϕ is normal, h is a constant of modulus 1 and in the cases for which T ϕ is known to be subnormal but not normal, h is a constant of modulus less than 1. It is of some interest to investigate the uniqueness of the functions h that relate f and g. Suppose h 1 and h 2 are in H and c 1 +T h1 f = g = c 2 +T h2 f. This is possible if and only if T z T h1 f = T z T h2 f, that is, if and only if T zh1 zh 2 f =0. Thus, f must be in (zχh 2 ) where χ is the inner factor of h 1 h 2. If f is not in any such subspace, the corresponding function h must be unique for every g. On the other hand, if χ is an inner function such that f is in (zχh 2 ) and c 1 + T h1 f = g, then for any h 3 in H and h 2 = h 1 + zχh 3, 4

5 it follows that g = c 2 + T h2 f for some constant c 2. In [6], the author made the following generalization of the set of g in H 2 for which T f+g is hyponormal. Definition Let H = {v H : v(0) = 0 and v 2 1}. Forf in H 2,let G f denote the set of g in H 2 such that for every u in H 2, sup <uv 0,g> sup <uv 0,f> v 0 H v 0 H To see how this definition is relevant to our work, note that if f is in H and u is in H 2, then by equation (1), sup <uv 0,f> = H f u. v 0 H Thus, when f and g are bounded analytic, T f+g is hyponormal if and only if g is in G f. For f in H 2, not necessarily the analytic part of a function in L,ifwe regard H f as a bounded operator from H into H 2, then we may proceed exactly as above to prove the following theorem. Theorem 2 If f and g are in H 2,theng is in G f if and only if g = c + T h f. for some constant c and some function h in H ( D) with h 1. We can now easily answer Question 1 of [6]. Corollary 3 For f in H 2, the following hold. (1) f is in G f. (2) If g is in G f,theng + λ is in G f for all complex numbers λ. (3) G f is balanced and convex; that is, if g 1 and g 2 are in G f and s 1 + s 2 1, thens 1 g 1 + s 2 g 2 is also in G f. (4) G f is weakly closed. (5) T χ G f G f for every inner function χ. Conversely, if G is a set that satisfies properties (1) to (5), then G G f. Proof. That G f has the indicated properties is Theorem 12 of [6]. To prove the converse statement, note that f is in G and by (3), (4), and (5), G contains T h f whenever h is in the weakly closed convex hull of the set of inner functions. By a theorem of Marshall [11, Corollary, page 496], the norm closed convex hull of the Blaschke products in H is the unit ball of H. Property (2) and Theorem 2 now imply the desired inclusion. 5

6 References [1] M. B. Abrahamse. Subnormal Toeplitz operators and functions of bounded type, Duke Math. J. 43(1976), [2] I. Amemiya, T. Ito, and T. K. Wong. On quasinormal Toeplitz operators, Proc. Amer. Math. Soc. 50(1975), [3] A. Brown and P. R. Halmos. Algebraic properties of Toeplitz operators, J. reine angew. Math. 213( ), [4] C. C. Cowen and J. J. Long. Some subnormal Toeplitz operators, J. reine angew. Math. 351(1984), [5] C. C. Cowen. More subnormal Toeplitz operators, J. reine angew. Math. 367(1986), [6] C. C. Cowen. Hyponormal and subnormal Toeplitz operators, to appear in Surveys of Recent Results in Operator Theory, Vol. 1 edited by J. B. Conway and B. B. Morrel. [7] P. R. Halmos. Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76(1970), [8] P. R. Halmos. Ten years in Hilbert space, Integral Equations and Operator Theory 2(1979), [9] T. Ito and T. K. Wong. Subnormality and quasinormality of Toeplitz operators, Proc. Amer. Math. Soc. 34(1972), [10] J. J. Long. Hyponormal Toeplitz Operators and Weighted Shifts, Thesis, Michigan State University, [11] D. E. Marshall. Blaschke products generate H, Bull. Amer. Math. Soc. 82(1976), [12] S. C. Power. Hankel Operators on Hilbert Space, Pitman, Boston, [13] D. Sarason. Generalized interpolation in H, Trans. Amer. Math. Soc. 127(1967), [14] Sun Shunhua. Bergman shift is not unitarily equivalent to a Toeplitz operator, Kexue Tongbao 28(1983),

7 [15] Sun Shunhua. On Toeplitz operators in the Θ-class, Scientia Sinica (Series A) 28(1985), Purdue University West Lafayette, Indiana